Find the point where line which passes through point `(1,2,3)` and is parallel to line ` vec r= hat i+ hat j+2 hat k+lambda( hat i-2 hat j+3 hat k)` meets the xy-plane.

Find the equation of a line passing through a point (2, -1, 3) and parallel to the line vec r=( hat i+ hat j)+lambda(2 hat i+ hat j-2 hat k)dot

Find the vector equation of the line passing through the point (1,2,3) and parallel to the planes vec rdot( hat i- hat j+2 hat k)=5 and vec rdot(3 hat i+ hat j+ hat k)=6.

Find the vector equation of the line passing through (1,2,3) and parallel to the planes vec rdot( hat i- hat j+2 hat k)a n d vec rdot(3 hat i+ hat j+ hat k)=6.

Find the vector and Cartesian forms of the equation of the plane passing through the point (1,\ 2,\ -4) and parallel to the lines vec r=( hat i+2 hat j-4 hat k)+lambda(2 hat i+3 hat j+6 hat k)a n d\ vec r=( hat i-3 hat j+5 hat k)+mu( hat i+ hat j- hat k)dot Also find the distance of the point (9, -8, -10) from the plane thus obtained.

Find the equation of the plane passing through (a ,b ,c) and parallel to the plane vec r is dot( hat i+ hat j+ hat k)=2.

Find the vector and cartesian equations of the line passing through the point P\ (1,\ 2,\ 3)\ and parallel to the planes vec rdot( hat i-\ hat j+2 hat k)=5\ and vec rdot(3 hat i+ hat j+ hat k)=6.

Find the vector equation of the line passing through (1, 2, 3 ) and parallel to the planes vec rdot( (hat i- hat j+2 hat k)=5\ a n d\ vec rdot((3 hat i+ hat j+ hat k)=6.

Find the vector and Cartesian equations of the line through the point (1, 2, 4) and perpendicular to the two lines vec r=(8 hat i-19 hat j+10 hat k)+lambda(3 hat i-16 hat j+7 hat k) and vec r=(15 hat i+29 hat j+5 hat k)+mu(3 hat i+8 hat j-5 hat k)dot

Find the vector equation of the line passing through (2,1,-1) and parallel to the line vec r= (hat i+hat j)+lambda(2hat i-hat j+hat k) . Also, find the distance between these two lines.

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes vec rdot(( hat i- hat j+2 hat k)=5\ a n d\ vec rdot((3 hat i+ hat j+ hat k)=6.